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Choosing problems for grad. students

I am coming to the point in my career where I will be expected to take graduate students, and I’d like some advice about finding problems for them. How responsible am I for making sure that a problem is solvable and not already under attack elsewhere? I have a (private) list of problems that might be suitable for attack with tropical methods, or using cluster algebras. In most cases, the reason that I have not worked on these problems myself is that I would have to do a fair bit of research to find out the current state of the field and make sure that I wasn’t missing something stupid. Is it fair to pass this sort of thing off to a graduate student?

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28 thoughts on “Choosing problems for grad. students”

Fair or not, my wife and I both started new projects as graduate students, so that’s was one of the major parts of the work. I figured it was simply the trade-off for getting to start a project, as opposed to picking up where another person left off.

In addition, I think that, from a skills standpoint, researching the state of a field/problem is something that any lifelong learner and researcher will have to confront eventually. The question is therefore not whether it should be passed off to graduate students or not, but *when* in a graduate student’s career it should be passed off.

A person I know who has been a very successful advisor recommends the following, which meshes with my limited experience (my first student is graduating this year):

Start the student out with a problem that you think is fairly easy and is something you could sit down and solve if you had sufficient motivation.

The main reason for this approach is that solving any reasonable problem and going through the process of writing it up greatly increases the student’s mathematical maturity. Even if the problem was fairly simple, afterwords the student thinks and works much more like a postdoc, and thus is then in a excellent position do something more substantial. (If it fact, the problem you though was “easy” doesn’t turn out to be much more difficult than expected, and ends up being a solid thesis in and of itself…)

I second Dunfeld’s suggestions. Also, while the student should be able (tih your help) to research the _published_ state of the art, you may want to send a couple of emails to relevant scholars. Just in case.

There is also the question of how much choosing the advisor should be doing. Grad students are autonomous beings, and like a parent, part of being a good advisor is not making your students too dependent on you.

Dunfield’s suggestion looks sound. Norman Steenrod recommended (according to the Automathography of Halmos) to point out an interesting “area of ignorance” rather than giving the student a specific problem. Also, some further tips can be found in this article written from the student’s perspective.

Choosing problems for grad students and giving career
advice are two sides of the same coin. I’ve observed and
participated in the process in many kinds of departments
(Cornell, Yale, Oregon, NYU-Courant, Columbia, UMass
in its early and recent years). So I could write a long list
of things to do or not do, but will refrain. To me the
primary question is always: Who is this student (strengths
and weaknesses, real career goals, background and
motivation)? Even in strong Ph.D. departments there is a
wide spectrum. Some students from Harvard and MIT
wind up outside academia, or teach at mid-level liberal
arts colleges, 5-year state colleges, etc. What would
make the student happy and is the student aiming too
high or low? (That’s all apart from the job market.)

The corollary is that a thesis problem has to be somewhat
tailored to the student, but with a knowledge of what
the real world demands. In many departments students
identify the easier and tougher advisors, thereby saving
you some trouble. When I was a student at Yale, Ore had
the reputation of being fairly easy; Mostow was not easy,
thus had few real Ph.D. students in spite of being an
exceptional mathematician. (I was one of the complicated
ones who thought he would teach at a cozy liberal arts
college but never did.) Some of your students will
probably wind up happily in non-academic jobs rather
than settle for teaching 3 sections of calculus each term.
Good luck, whichever side of the fence you are on.

Well, senior guys who have been successful at guiding students seem to have different approaches.

I’ve found Ravi Vakil’s page, Allen Kuntson’s page (who does say “The advisor should know what other people are working on. Few experiences are more frustrating than working for years on a problem, to have someone else solve it; one of them is to find out that it was already solved long ago! “), and I also saw this quote of V.I. Arnold “I never assign a thesis topic to my students. This is like assigning them a spouse. I merely show them what is known and unknown”.

If I were to advise a student, I’d force him/her to write a small paper first within 3 months (like fleshing-out a proof not detailed in a big paper plus finding one new corollary from it with little effort), just to get things started, and only then set him/her free to roam .

My grandfather-in-law apparently said the following: a good master’s thesis problem was one he could solve in an afternoon, and a good PhD thesis problem was one for which he could come up with several reasonable approaches in a week.

In Sep. of 2007 my supervisor suggested that I start reading a series of papers by mathematician A which was about 600 page of math. He told me that A was “confused” and that it would be easy for me to find improvements and essentially rewrite his work in much more compact and elegant manner and maybe solve a problem that A was trying to solve for some time. After a year of reading A’s work I came to conclusion that A’s work was far from confused. After giving a couple of talks to my supervisor he agreed that this was uglier than he thought and that there was no clean unifying answer to A’s work. He then urged me to work on the aforementioned open problem (Sep. 2008). What happened next was that A announced that he had solved the problem in Jun. of 2009 and I am left with nothing and I have to graduate by Aug. 2010 because my supervisor wont support me after the end of my fourth year.

To PhD student: sorry to hear that, your situation is definitely not ideal, but not desperate. You have about 8 months left, be very organized (start LaTeXing regularly a lot of material if you haven’t already, like 2 days a week). You’re one of the few experts on the stuff A has been doing, and 600 pages is no small thing, so you’re not with nothing, you can use it to your advantage: try to find applications to other problems, or reprove a couple of steps differently, or extract from it results or methods of independent interest which can be developped a bit. I’d also advise you to keep one day per week to work on small side projects on fairly different topics in your area: you can always add that as an extra chapter to your thesis to show you have broader interests, and you could even manage that way to get a short paper by march or april and submit it to a quick-publication journal like Proc AMS… It will be a few fast-paced months, but it is manageable, don’t despair (I know of similar cases with a happy ending).

It is hard not to feel desperate and angry. I don’t have much time in the next month because I am teaching this semester. Even if I finish by Aug 2010 I need to apply for postdocs “now”. And I have nothing to show for my 3 years as a PhD student, I mean I would not hire myself! I have lost my appetite for math I hope I get my degree and get a job outside academia (banks). The other option is getting a loan and staying for the 5th year on my own money.

Standard advice for PhD students in the North American system: if you are moving on to a career for which having the actual PhD degree is of little benefit, you should finish the thesis (assuming you do want to finish), but delay actually filing and getting your PhD as long as regulations reasonably allow. This way if you want back in academia 2 years later you can still qualify for postdoctoral positions which require you to be a recent PhD.

To PhD student.. there might be something salvageable here. If you solve this problem or a related problem without seeing his proof, and your alternate proof is truly different enough, you may be able to graduate anyhow. Of course the end result is still far from ideal, but keep in mind it is actually pretty common for people in some fields to work on the same problem using similar methods, and come up with competing proofs for the same result. If you already have seen his argument however then you’re probably screwed.

If you solve this problem or a related problem without seeing his proof

Once the first solution has been disseminated, you can’t claim to have solved it independently just because you didn’t read it, so that’s not relevant (although a genuinely different proof would of course be a valuable contribution). However, there are probably still opportunities here. When someone makes a breakthrough, there’s usually a lot of valuable follow-up work, and disappointed competitors are often in a good position to participate in that. It may be small potatoes compared to the original open problem, but that was probably pretty ambitious for a thesis problem, considering that mathematician A spent years on it. You can write a solid but not great thesis, and it won’t keep you from doing great work later.

Your advisor ought to be supportive. Giving up and saying “Oh well, there goes your career” isn’t acceptable – if that’s how he/she has reacted, then your hope for a career in mathematics depends on finding another mentor who can help. However, a supportive advisor can help a lot in putting your job application in the best light. You are working in a hot area, you have mastered an intimidating body of work, you were working on a plausible approach to an important conjecture, but then you got scooped. Nevertheless, you are currently working on applying the new technique to some related problems, and you are in a great position to take advantage of it, so you have a very bright future. A letter like this isn’t going to get you a job at the top schools, which get so many great candidates that they don’t have to take chances on unusual cases, but not all hope is lost.

I disagree with the statement that the important thing is whether or not you have seen the other solution. If I were looking at your application, I would want to see whether you had done something new with the material you had absorbed; either found a different proof, or a new application. For that matter, if you visited campus and gave me a great explanation of this field, even without mentioning any of your own results, I would consider this in your favor; I find that, in any hot field, there are very few people who understand and can explain it well.

But everyone here is giving you advice on the assumption that you want to go into academia, and you’ve said you don’t. There’s nothing wrong with wanting to apply your knowledge in a bank, as long as its what you want and not what you feel you are forced into. If any readers have advice on how to get a good nonacademic job, that might be more useful to you.

Just to clarify.. I meant that if the proof is different enough it can still have some merit. If you’ve seen the proof and your methods are similar, I just think it would be pretty hard to come up with a noticeably different proof, not that the standard for a proof has to do with what the author has seen.

Thank you all for the feedback. Reading my own post I see how it could have been misleading, I am not set on leaving academia but I can’t stay in academia if I finish this year because as I said then I have to apply now and applying now looks like a joke to me. If I stay, I have to take a loan for my 5th year which is entirely possible.

@PhD student: I wish I knew. But you have time while you’re not on the job market to start picking up some outside skills.

If you’re interested in working at a bank, start reading about financial mathematics and write a short paper about it, especially if you can explain some mathematically interesting structure to a non-mathematical audience. It doesn’t even have to be particularly original. Then when you apply for banking jobs you can say, “look! I know about financial mathematics, and I even have a short paper you might be interested in to prove it.”

The catch is: what if you don’t know what non-academic field you want to work in?

David Speyer asked: “How responsible am I for making sure that a problem is solvable and not already under attack elsewhere?”

You are completely responsible for this! Most students don’t have the experience or know-how to judge these issues – and the most ambitious smart-seeming students are often the ones who, left to their own devices, will choose overly difficult projects and forget to search the literature.

You are also responsible for teaching them how to write a good math paper, how to publish a math paper, how to give a good talk, how to find conferences to give talks at, and how to get a job.

My Ph.D. was two rather hard (one of them more than a century old) technical problems. Both problems were closures, so I spent most of my first post-doc learning new material, and trying to figure out what to work on.

If you have a student working on a tropical problem, then he’ll learn tropical geometry at the very least and that’s a rich area right now with lots of open problems. Even if his thesis problem gets scooped and all he has is an independent proof (similar to PhD student), he will still have learned different tools to use on other problems, so he won’t be left with no direction. Also, he can probably find similar problems that will have similar solutions once he delves into the area. Similarly for cluster algebras, and doubly so for working on clusters via tropical geometry.

Here are some random thoughts about looking for a non-academic job. It can be a lot of work, and unfortunately there aren’t any easy answers.

1. Try to appear confident, and definitely don’t let potential employers think you can’t get an academic job and view them as a fall-back option. Psychologically, people want something more if they think other people want it, and even though there are plenty of reasons why an academic career might not work out that don’t reflect poorly on the applicant, they’ll be suspicious that you might be hiding something. If you are hesitant or apologetic, or seem desperate, or denigrate your own work or academic job chances, or explain that you lack background in this new field since you never thought you might have to leave academia, it will hurt your chances.

2. Demonstrate real enthusiasm for whatever area you are applying in, and try to learn as much as you can before you interview. (Read books, search the web, talk with contacts in the field, etc.) Getting a Ph.D. in math means you are smart and diligent, but employers may fear that you will take a long time getting up to speed in a new area, discover in the process that you don’t like it, and never really get on track, instead spending years feeling bitter and wishing you were doing something else. You need to make it clear that you know what you are getting into and have already started building expertise.

3. Don’t just rely on close friends and colleagues for advice and contacts. They are probably too close to you in background to make this effective. Reach out to “weak ties” (people you knew in college but haven’t had a lot of contact with, people who overlapped briefly with you in grad school before leaving for industry, people you met at parties, friends of friends, etc.). Any individual probably won’t help you much, but you’re maximizing the chances of getting a totally different perspective or opportunity.

4. Try to identify people in companies you might like to work for, and ask to do an “informational interview” well in advance of when you will be applying for jobs. This is a brief conversation (in person, or by phone if you are in a different city) about what they are looking for when hiring, what’s going on at their company and in their field, etc. Of course, this is a transparent attempt to get their attention for hiring, but it is also a good way to gather information. Ask for fifteen or at most thirty minutes, and nobody will be offended (they may well be too busy and decline, but they may also feel flattered); when the time is up, let them continue but don’t prolong the conversation yourself. Don’t ask the CEO or other executives; instead, ask someone who is doing the job you’d like to be doing in five years, or someone who is managing a small group of people (maybe ten or fewer).

5. When you do an informational interview, come well prepared: know what their company does, what they do at it, recent news in the industry, etc. Ask good questions, especially if there are any pauses in the conversation, and don’t talk a lot about yourself except in response to their questions. (You want to mention briefly at the start what your background is, but the idea is that you are learning what they do and want, rather than selling yourself.) If they suggest additional background you would need to do the job, it gives you a good opportunity to start developing that background and then refer to it when you apply for an actual job.

6. Philosophically, you should take the approach of figuring out what employers want and making the best case you honestly can that you will give it to them. This is not a very academic attitude, and mathematicians are often tempted to tell people what they should want, instead. Changing people’s minds is a good long-term project, but it rarely works in the short term. When you want to convince someone of something quickly, instead of focusing on the reasons you consider most compelling (but which they may not appreciate or value), you should try to identify what they do value and see how strong a case you can make in those terms. If you can’t make such a case, then you’re probably not going to get anywhere quickly, and if you can, then that’s enough, even if you think you could have made an even stronger case to a different audience.

7. Don’t feel embarrassed. There’s a myth in academia that if you are smart enough and work hard enough, your career will automatically sort itself out. That’s not true, but it can lead people to feel like a failure when they have to devote time to figuring out a career path. Fortunately, outside of academia nobody considers this problematic or embarrassing.

Have you tried talking to Mathematician A? He might have some good advice for you.

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